Example 4.4 Let \((M,\mathcal {F})\)  be a complete transverse parallel foliated connected manifold (see for instance [1]} or [2]}). Let \(p:\widetilde{M}\rightarrow M\)  denote the universal covering map and consider the induced foliation \(\widetilde{\mathcal {F}}\)  on \(\widetilde{M}\) . This foliation is simple so that we have that \(X=\widetilde{M}/\widetilde{\mathcal {F}}\)  is a Hausdorff manifold and the canonical projection \(\pi _{\textnormal {bas}}:\widetilde{M}\rightarrow X\)  is a surjective submersion. It turns out that with this data it is possible to obtain a natural structure of principal groupoid bi-bundle \((\widetilde{M},p,\pi _{\textnormal {bas}}):\textnormal {Hol}(M,\mathcal {F}){} \pi _1(M)\ltimes X\)  between the holonomy groupoid \(\textnormal {Hol}(M,\mathcal {F})\rightrightarrows M\)  and the action groupoid \(\pi _1(M)\ltimes X \rightrightarrows X\) , that is, a Morita equivalence. Therefore, if \(M\)  has finite fundamental group then as a consequence of Wasserman's result and Proposition REF  we get that there exist Morse Lie groupoid morphisms on \(\textnormal {Hol}(M,\mathcal {F})\rightrightarrows M\) .

